probabilistic-robotics/metropolis-hastings-algos.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a nice discussion that is intuitive and easy to follow on Metropolis-Hastings Algorithm on [Wikipedia](https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm). [Here](http://galton.uchicago.edu/~eichler/stat24600/Handouts/l12.pdf) are some more nice notes on MH. Simulated Annealing, a method inspired my Metropolis-Hastings Algorithm is also discussed here.\n",
    "\n",
    "The pseudo code for Metropolis-Hastings Algorithm is discussed below --\n",
    "\n",
    "* Initialize $x_0$.\n",
    "    * For $i = 1 \\text{ to } N-1$\n",
    "        * Sample $u \\sim \\mathcal{U}_{[0, 1]}$\n",
    "        * Sample $x^{*} \\sim q(x|x^{(i)})$\n",
    "        * If $u < \\mathcal{A}(x^(i), x^*) = \\min\\{\\frac{p(x^*)q(x^{(i)}|x^*)}{p(x^{(i)})q(x^*|x^{(i)})}\\}$\n",
    "            * $x^{i+1} = x^{*}$\n",
    "        * else\n",
    "            * $x^{i+1} = x^{(i)}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Import the required modules\n",
    "from scipy.stats import norm\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expected Value of x  = 4.01284026674 with variance = 11.4278689915\n"
     ]
    },
    {
     "data": {
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YVxlUq9Wm09YDfHlL8tVOZ6CLVDudgQ7at9dsK98RG65dJfqFwOaI2BoRu4DV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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8d00ddd550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Let's sample a random distribution and plot the histogram of the samples\n",
    "x = np.hstack((np.random.normal(7, 1, 10000), np.random.normal(1, 2, 10000)))\n",
    "print('Expected Value of x  = {} with variance = {}'.format(np.mean(x), np.var(x)))\n",
    "plt.hist(x, bins=100)\n",
    "plt.title('Samples drawn from a Gaussian Distribution')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# This function returns the unnormalized probabilites \n",
    "def p(x):\n",
    "    return (norm(7, 1).pdf(x) + norm(1, 2).pdf(x))/2\n",
    "\n",
    "# Initialize x_0\n",
    "x = 5\n",
    "# Number of samples\n",
    "N = 10000\n",
    "# I think when we are not sure\n",
    "# we should keep the `sigma` as large\n",
    "# as possible, so that we sample the entire space\n",
    "sigma = 100\n",
    "# List of sampled points\n",
    "x_sampled = []\n",
    "for i in range(N-1):\n",
    "    # `u` lies in [0, 1]\n",
    "    u = np.random.uniform()\n",
    "    # Sample `x_star` from a gaussian distribution centered around `x`\n",
    "    x_star = norm(x, sigma).rvs()\n",
    "    if u < min(1, (p(x_star)*norm(x_star, sigma).pdf(x))/(p(x)*norm(x, sigma).pdf(x_star))):\n",
    "        x = x_star\n",
    "    x_sampled.append(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Expected Value of x  = 4.08718070983 with variance = 11.2499887046\n"
     ]
    },
    {
     "data": {
      "image/png": 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PqJnAq4CPRcTdkv6OVO2wpNywxpK0L6mUvAB4Crhc0gd66PvT9Me+44k+It4+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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8d00ddd4d0>"
      ]
     },
     "metadata": {},
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   ],
   "source": [
    "# Plot the sampled distribution\n",
    "print('Expected Value of x  = {} with variance = {}'.format(np.mean(x_sampled), np.var(x_sampled)))\n",
    "plt.hist(x_sampled, bins=100)\n",
    "plt.title('Samples drawn from a Gaussian Distribution')\n",
    "plt.show()"
   ]
  }
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